PreCalculus Exercises Section 5.3

19) If in Figure 1, what are the values of the other quantities.
Solution:
Because and are complimentary angles = 90 - 17.8 = 72.2 degrees.
Note that 0.2 of a degree is (.2)(60) = 12.0 minutes. So = 72 degrees and 12 minutes. Which would normally be written as .

We can now use the definition of the sine function ( ) to obtain
so that. Note that the value of a could have been determined by using .
The Pythagorean Theorem implies that
Observe that if we use the sine function to determine b we obtain
The discrepency is due to round-off error.

See your author's comments at the end of Example 2 on page 412.

24) If in Figure 1, what are the values of the other quantities.
Solution:
Because and are complimentary angles = 90 - 54 = 36 degrees.
We can now use the sine function to obtain from which it follows that
We can now use the cosine function to obtain from which it follows that

28) If in Figure 1, a = 22 and b = 46.2 what are the values of the other quantities.
Solution:
The Pythagorean Formula yields
At this point it is possible to work with any one of sine, cosine, or tangent functions. I choose the sine function.
The sine function yields .

Properties of inverse functions (in particular the inverse of the sine function) may be employed to obtain

Had I decided to use the cosine function, the following would have been the solution:
The cosine funtion yields
Properties of inverse functions (in particular the inverse of the cosine function) may be employed to obtain

Had I decided to use the tangent function, the following would have been the solution:
The tangent funtion yields
Properties of inverse functions (in particular the inverse of the tangent function) may be employed to obtain

45) Find the height of a tree (growing on level ground) if at a point 105 feet from the base of the tree the angle to its top relative to the horizontal is 65.3 degrees.
Solution:
The picture at the right models the question.
The tangent function is ideal for determining the height of the tree because tan(65.3) = h/105 so that h = (105)tan(65.3) =228 feet.

45) With reference to the diagram at the right, show that
Solution:
Note that and and therefore

which is what was to be demonstrated.